Most people who run fixed do it for fun or training purposes. It is fun, and from a training perspective it helps develop smoothness and spin. As for practical advantages they are few.

I do find I can climb in a taller gear fixed than free, because my pedal stroke is more completely circular. this also helps you go fast, but is more difficult to measure that advantage.

the other advantage is durability, Freewheels have a lot of fragile moving parts that can wear or get gummed up (especially bad in winter) A fixed cog properly installed will give many more miles of trouble free service.

These advantages are generally not enough in and of themselves to outweigh the convenience of coasting and shifting. Unless of course you just really love riding fixed. Which is why most of us here do it.